Let \(\mathbf {p}=[st_1, st_2, st_3, x^2, xy, xz, y^2, yz, z^2, x, y, z, 1]^T\) .
Writing the equations as a matrix-vector multiply \(\mathsf {M}\mathbf {p}=0\) , we first eliminate \(s\mathbf {t}\)  using Gauss-Jordan elimination. This leaves three quadratic equations in ten monomials on \(x,y,z\)  – a 3Q3 problem. Kukelova et al. [1]} introduced a fast solver for the 3Q3 problem which produces up to eight solutions for \(x,y,z\) . We used the implementation by Larssonhttps://github.com/vlarsson/re3q3 which uses several techniques to improve the stability of the original solver [2]}. One we have obtained solutions for \(x,y,z\) , we solve for \(\mathbf {t}\)  by backsubstitution into the original equations.
